Bases and dimensions of bivariate hierarchical tensor-product splines

We prove that the dimension of bivariate tensor-product spline spaces of bi-degree (d,d)(d,d) with maximum order of smoothness on a multi-cell domain (more precisely, on a set of cells from a tensor-product grid) is equal to the number of tensor-product B-spline basis functions, defined by only single knots in both directions, acting on the considered domain. A certain reasonable assumption on the configuration of the cells is required.This result is then generalized to the case of piecewise polynomial spaces, with the same smoothness properties mentioned above, defined on a multi-grid multi-cell domain (more precisely, on a set of cells from a hierarchy of tensor-product grids). Again, a certain reasonable assumption regarding the configuration of cells is needed.Finally, it is observed that this construction corresponds to the classical definition of hierarchical B-spline bases. This allows to conclude that this basis spans the full space of spline functions on multi-grid multi-cell domains under reasonable assumptions.